Nonlinear optical response of resonantly driven polaron-polaritons

Exciton polaritons in two-dimensional semiconductors inside microcavities are powerful platforms to explore hybrid light-matter quantum systems. Here, we study a macroscopic coherent population of the lowest energy state of polaron-polaritons, which are quasiparticles formed by the dressing of exciton polaritons by particle-hole excitations in a surrounding electron gas. Using a nonperturbative many-body theory to describe exciton-electron correlations combined with a nonequilibrium theory for the macroscopically populated state, we show that the electrons strongly affect the collective properties of the polaron-polaritons. This stems from the dependence of the polaron-polariton energy and the interaction between them on the electron density, which leads to strong nonlinearities. We identify stable and unstable regimes of the polaron-polaritons by calculating its excitation spectrum, and show that they result in prominent hysteresis effects when the electron density is varied. Our results should be readily observable using present experimental technology.

Figure 1

(a) Schematic diagram illustrating the different physical regimes realised by polaron-polaritons in a 2DEG. The arrow represents our approach to analyze the Bose-Fermi mixture, using the polaron-polaritons as our starting point. (b) Condensate density $n_0$ as a function of pump laser amplitude $F_{\text{pu}}$ for the pump frequency ${\epsilon }_{\text{pu}}/2\mathrm{\Omega }=-2.22$ and various electron densities $n_e$. (c) Condensate density as a function of electron density $n_e$ for fixed pump laser amplitude indicated by the vertical line in (b). Blue solid (yellow dashed) lines indicate stable (unstable) configurations.

Figure 2

(a) Spectral density of a single cavity photon. White line is the bare photon energy and black lines are the bare upper and lower polariton dispersions. The energy ${\epsilon }_0$ of the lowest polaron-polariton branch at $\delta /2\mathrm{\Omega }=-2.11$ is indicated by a red star (b) The energy ${\epsilon }_0$ as a function of the electron density. The pump frequency at ${\epsilon }_{\text{pu}}/2\mathrm{\Omega }=-2.22$ is shown as a horizontal line. (c) The interaction between two polaron-polaritons as a function of the electron density.

Figure 3

Bogoliubov dispersions $E_{\mathbf{k}}$ as a function of the in-plane momentum $k=\left|\mathbf{k}\right|$ for the three different cases marked by the triangles in Fig. 1. Panels (a), (c), and (e) show $\mathrm{Re}\left[E_{\mathbf{k}}\right]$ and panels (b), (d), and (f) show $\mathrm{Im}\left[E_{\mathbf{k}}\right]$. Momenta are expressed in the units of $k_r=\sqrt{4\pi n_0}$. The stability of different cases can be deduced from the sign of the imaginary part of $E_{\mathbf{k}}$.